One Round DKG (#589)

* Upstream GBP, divisor, circuit abstraction, and EC gadgets from FCMP++

* Initial eVRF implementation

Not quite done yet. It needs to communicate the resulting points and proofs to
extract them from the Pedersen Commitments in order to return those, and then
be tested.

* Add the openings of the PCs to the eVRF as necessary

* Add implementation of secq256k1

* Make DKG Encryption a bit more flexible

No longer requires the use of an EncryptionKeyMessage, and allows pre-defined
keys for encryption.

* Make NUM_BITS an argument for the field macro

* Have the eVRF take a Zeroizing private key

* Initial eVRF-based DKG

* Add embedwards25519 curve

* Inline the eVRF into the DKG library

Due to how we're handling share encryption, we'd either need two circuits or to
dedicate this circuit to the DKG. The latter makes sense at this time.

* Add documentation to the eVRF-based DKG

* Add paragraph claiming robustness

* Update to the new eVRF proof

* Finish routing the eVRF functionality

Still needs errors and serialization, along with a few other TODOs.

* Add initial eVRF DKG test

* Improve eVRF DKG

Updates how we calculcate verification shares, improves performance when
extracting multiple sets of keys, and adds more to the test for it.

* Start using a proper error for the eVRF DKG

* Resolve various TODOs

Supports recovering multiple key shares from the eVRF DKG.

Inlines two loops to save 2**16 iterations.

Adds support for creating a constant time representation of scalars < NUM_BITS.

* Ban zero ECDH keys, document non-zero requirements

* Implement eVRF traits, all the way up to the DKG, for secp256k1/ed25519

* Add Ristretto eVRF trait impls

* Support participating multiple times in the eVRF DKG

* Only participate once per key, not once per key share

* Rewrite processor key-gen around the eVRF DKG

Still a WIP.

* Finish routing the new key gen in the processor

Doesn't touch the tests, coordinator, nor Substrate yet.
`cargo +nightly fmt && cargo +nightly-2024-07-01 clippy --all-features -p serai-processor`
does pass.

* Deduplicate and better document in processor key_gen

* Update serai-processor tests to the new key gen

* Correct amount of yx coefficients, get processor key gen test to pass

* Add embedded elliptic curve keys to Substrate

* Update processor key gen tests to the eVRF DKG

* Have set_keys take signature_participants, not removed_participants

Now no one is removed from the DKG. Only `t` people publish the key however.

Uses a BitVec for an efficient encoding of the participants.

* Update the coordinator binary for the new DKG

This does not yet update any tests.

* Add sensible Debug to key_gen::[Processor, Coordinator]Message

* Have the DKG explicitly declare how to interpolate its shares

Removes the hack for MuSig where we multiply keys by the inverse of their
lagrange interpolation factor.

* Replace Interpolation::None with Interpolation::Constant

Allows the MuSig DKG to keep the secret share as the original private key,
enabling deriving FROST nonces consistently regardless of the MuSig context.

* Get coordinator tests to pass

* Update spec to the new DKG

* Get clippy to pass across the repo

* cargo machete

* Add an extra sleep to ensure expected ordering of `Participation`s

* Update orchestration

* Remove bad panic in coordinator

It expected ConfirmationShare to be n-of-n, not t-of-n.

* Improve documentation on  functions

* Update TX size limit

We now no longer have to support the ridiculous case of having 49 DKG
participations within a 101-of-150 DKG. It does remain quite high due to
needing to _sign_ so many times. It'd may be optimal for parties with multiple
key shares to independently send their preprocesses/shares (despite the
overhead that'll cause with signatures and the transaction structure).

* Correct error in the Processor spec document

* Update a few comments in the validator-sets pallet

* Send/Recv Participation one at a time

Sending all, then attempting to receive all in an expected order, wasn't working
even with notable delays between sending messages. This points to the mempool
not working as expected...

* Correct ThresholdKeys serialization in modular-frost test

* Updating existing TX size limit test for the new DKG parameters

* Increase time allowed for the DKG on the GH CI

* Correct construction of signature_participants in serai-client tests

Fault identified by akil.

* Further contextualize DkgConfirmer by ValidatorSet

Caught by a safety check we wouldn't reuse preprocesses across messages. That
raises the question of we were prior reusing preprocesses (reusing keys)?
Except that'd have caused a variety of signing failures (suggesting we had some
staggered timing avoiding it in practice but yes, this was possible in theory).

* Add necessary calls to set_embedded_elliptic_curve_key in coordinator set rotation tests

* Correct shimmed setting of a secq256k1 key

* cargo fmt

* Don't use `[0; 32]` for the embedded keys in the coordinator rotation test

The key_gen function expects the random values already decided.

* Big-endian secq256k1 scalars

Also restores the prior, safer, Encryption::register function.

crypto/dkg/src/lib.rs

@@ -21,6 +21,10 @@ pub mod encryption;
 #[cfg(feature = "std")]
 pub mod pedpop;
 
+/// The one-round DKG described in the [eVRF paper](https://eprint.iacr.org/2024/397).
+#[cfg(all(feature = "std", feature = "evrf"))]
+pub mod evrf;
+
 /// Promote keys between ciphersuites.
 #[cfg(feature = "std")]
 pub mod promote;
@@ -205,25 +209,37 @@ mod lib {
     }
   }
 
-  /// Calculate the lagrange coefficient for a signing set.
-  pub fn lagrange<F: PrimeField>(i: Participant, included: &[Participant]) -> F {
-    let i_f = F::from(u64::from(u16::from(i)));
+  #[derive(Clone, PartialEq, Eq, Debug, Zeroize)]
+  pub(crate) enum Interpolation<F: Zeroize + PrimeField> {
+    Constant(Vec<F>),
+    Lagrange,
+  }
 
-    let mut num = F::ONE;
-    let mut denom = F::ONE;
-    for l in included {
-      if i == *l {
-        continue;
+  impl<F: Zeroize + PrimeField> Interpolation<F> {
+    pub(crate) fn interpolation_factor(&self, i: Participant, included: &[Participant]) -> F {
+      match self {
+        Interpolation::Constant(c) => c[usize::from(u16::from(i) - 1)],
+        Interpolation::Lagrange => {
+          let i_f = F::from(u64::from(u16::from(i)));
+
+          let mut num = F::ONE;
+          let mut denom = F::ONE;
+          for l in included {
+            if i == *l {
+              continue;
+            }
+
+            let share = F::from(u64::from(u16::from(*l)));
+            num *= share;
+            denom *= share - i_f;
+          }
+
+          // Safe as this will only be 0 if we're part of the above loop
+          // (which we have an if case to avoid)
+          num * denom.invert().unwrap()
+        }
       }
-
-      let share = F::from(u64::from(u16::from(*l)));
-      num *= share;
-      denom *= share - i_f;
     }
-
-    // Safe as this will only be 0 if we're part of the above loop
-    // (which we have an if case to avoid)
-    num * denom.invert().unwrap()
   }
 
   /// Keys and verification shares generated by a DKG.
@@ -232,6 +248,8 @@ mod lib {
   pub struct ThresholdCore<C: Ciphersuite> {
     /// Threshold Parameters.
     pub(crate) params: ThresholdParams,
+    /// The interpolation method used.
+    pub(crate) interpolation: Interpolation<C::F>,
 
     /// Secret share key.
     pub(crate) secret_share: Zeroizing<C::F>,
@@ -246,6 +264,7 @@ mod lib {
       fmt
         .debug_struct("ThresholdCore")
         .field("params", &self.params)
+        .field("interpolation", &self.interpolation)
         .field("group_key", &self.group_key)
         .field("verification_shares", &self.verification_shares)
         .finish_non_exhaustive()
@@ -255,6 +274,7 @@ mod lib {
   impl<C: Ciphersuite> Zeroize for ThresholdCore<C> {
     fn zeroize(&mut self) {
       self.params.zeroize();
+      self.interpolation.zeroize();
       self.secret_share.zeroize();
       self.group_key.zeroize();
       for share in self.verification_shares.values_mut() {
@@ -266,16 +286,14 @@ mod lib {
   impl<C: Ciphersuite> ThresholdCore<C> {
     pub(crate) fn new(
       params: ThresholdParams,
+      interpolation: Interpolation<C::F>,
       secret_share: Zeroizing<C::F>,
       verification_shares: HashMap<Participant, C::G>,
     ) -> ThresholdCore<C> {
       let t = (1 ..= params.t()).map(Participant).collect::<Vec<_>>();
-      ThresholdCore {
-        params,
-        secret_share,
-        group_key: t.iter().map(|i| verification_shares[i] * lagrange::<C::F>(*i, &t)).sum(),
-        verification_shares,
-      }
+      let group_key =
+        t.iter().map(|i| verification_shares[i] * interpolation.interpolation_factor(*i, &t)).sum();
+      ThresholdCore { params, interpolation, secret_share, group_key, verification_shares }
     }
 
     /// Parameters for these keys.
@@ -304,6 +322,15 @@ mod lib {
       writer.write_all(&self.params.t.to_le_bytes())?;
       writer.write_all(&self.params.n.to_le_bytes())?;
       writer.write_all(&self.params.i.to_bytes())?;
+      match &self.interpolation {
+        Interpolation::Constant(c) => {
+          writer.write_all(&[0])?;
+          for c in c {
+            writer.write_all(c.to_repr().as_ref())?;
+          }
+        }
+        Interpolation::Lagrange => writer.write_all(&[1])?,
+      };
       let mut share_bytes = self.secret_share.to_repr();
       writer.write_all(share_bytes.as_ref())?;
       share_bytes.as_mut().zeroize();
@@ -352,6 +379,20 @@ mod lib {
         )
       };
 
+      let mut interpolation = [0];
+      reader.read_exact(&mut interpolation)?;
+      let interpolation = match interpolation[0] {
+        0 => Interpolation::Constant({
+          let mut res = Vec::with_capacity(usize::from(n));
+          for _ in 0 .. n {
+            res.push(C::read_F(reader)?);
+          }
+          res
+        }),
+        1 => Interpolation::Lagrange,
+        _ => Err(io::Error::other("invalid interpolation method"))?,
+      };
+
       let secret_share = Zeroizing::new(C::read_F(reader)?);
 
       let mut verification_shares = HashMap::new();
@@ -361,6 +402,7 @@ mod lib {
 
       Ok(ThresholdCore::new(
         ThresholdParams::new(t, n, i).map_err(|_| io::Error::other("invalid parameters"))?,
+        interpolation,
         secret_share,
         verification_shares,
       ))
@@ -383,6 +425,7 @@ mod lib {
   /// View of keys, interpolated and offset for usage.
   #[derive(Clone)]
   pub struct ThresholdView<C: Ciphersuite> {
+    interpolation: Interpolation<C::F>,
     offset: C::F,
     group_key: C::G,
     included: Vec<Participant>,
@@ -395,6 +438,7 @@ mod lib {
     fn fmt(&self, fmt: &mut fmt::Formatter<'_>) -> fmt::Result {
       fmt
         .debug_struct("ThresholdView")
+        .field("interpolation", &self.interpolation)
         .field("offset", &self.offset)
         .field("group_key", &self.group_key)
         .field("included", &self.included)
@@ -480,12 +524,13 @@ mod lib {
       included.sort();
 
       let mut secret_share = Zeroizing::new(
-        lagrange::<C::F>(self.params().i(), &included) * self.secret_share().deref(),
+        self.core.interpolation.interpolation_factor(self.params().i(), &included) *
+          self.secret_share().deref(),
       );
 
       let mut verification_shares = self.verification_shares();
       for (i, share) in &mut verification_shares {
-        *share *= lagrange::<C::F>(*i, &included);
+        *share *= self.core.interpolation.interpolation_factor(*i, &included);
       }
 
       // The offset is included by adding it to the participant with the lowest ID
@@ -496,6 +541,7 @@ mod lib {
       *verification_shares.get_mut(&included[0]).unwrap() += C::generator() * offset;
 
       Ok(ThresholdView {
+        interpolation: self.core.interpolation.clone(),
         offset,
         group_key: self.group_key(),
         secret_share,
@@ -528,6 +574,14 @@ mod lib {
       &self.included
     }
 
+    /// Return the interpolation factor for a signer.
+    pub fn interpolation_factor(&self, participant: Participant) -> Option<C::F> {
+      if !self.included.contains(&participant) {
+        None?
+      }
+      Some(self.interpolation.interpolation_factor(participant, &self.included))
+    }
+
     /// Return the interpolated, offset secret share.
     pub fn secret_share(&self) -> &Zeroizing<C::F> {
       &self.secret_share